Defect behavior during growth of heavily phosphorus doped Czochralski silicon crystals (II): Theoretical study

From the late 1970s to the early 2010s, many experimental studies have been conducted on the impact of dopants and impurities on intrinsic point defect behavior and grown-in defect formation during Si crystal growth.^1–8 The types of dopants that have been reported are p-type (B and Ga), neutral (C, Ge, and Sn), and n-type (P, As, Sb, and Bi). The impacts of impurities (H, C, O, and N) have also been investigated for the purpose of improving the quality of Si crystals.^4,9–15 Among the p-type and n-type dopants, B, P, and As can be doped at the highest concentration. The highest concentration of B, P, and As for which experimental results have been reported up until the early 2010s was about 1 × 10¹⁹ B cm⁻³, 3.5 × 10¹⁹ P cm⁻³ and 3 × 10¹⁹ As cm⁻³, respectively.⁴ Based on a model that quantitatively explains the intrinsic point defect behavior using density functional theory (DFT) calculations,¹⁶ the concentration distribution of intrinsic point defects valid for all pulling conditions in large-diameter Czochralski-Si (Cz-Si) crystal growth has been determined by computer simulation.^17,18 Theoretical studies showed that the self-interstitial (I) and vacancy (V) formation energies around dopant atoms change depending on the type and size of the dopants, i.e., the electrical state and the local strain around the dopants. That is, the Si crystal becomes I-rich by B doping up to 1 × 10¹⁹ B cm⁻³ while it becomes V-rich by P doping up to 3.5 × 10¹⁹ P cm⁻³and As doping up to 3 × 10¹⁹ As cm⁻³.

In the last decade, heavily P doped (HP-Cz-Si) crystals have been widely used in power devices. To reduce the power consumption, the crystal resistivity must be reduced as much as possible. Currently, the most advanced crystals can be manufactured with a resistivity as low as 0.6 mΩ cm (1.3 × 10²⁰ P cm⁻³). The defect behavior in CZ-Si has been reported to change significantly when the P concentration in the Si crystal exceeds 3 × 10¹⁹ P cm⁻³.^4,19–26 That is, the void formation is significantly suppressed with P doping over (3–4) × 10¹⁹ P cm⁻³.^4,19 Senda et al.²⁰ observed plate-like SiP²¹ precipitates of 100–200 nm in as-grown HP-CZ-Si crystals at around 8 × 10¹⁹ P cm⁻³. Zeng et al.^22,23 also reported the formation of oxygen precipitates from heterogeneous nuclei of small SiP precipitates during the heat treatment of HP-Cz-Si crystals. Subsequently, Wu et al.²⁴ also reported that post-annealing of HP-CZ-Si crystals with a maximum [P] of 7.35 × 10¹⁹ P cm⁻³ at 450–1050 °C resulted in the formation of SiP precipitates of various crystallographic morphologies with dependent of temperature. Voronkov et al.²⁵ and Nakamura et al.²⁶ claimed that the increase of interstitial P (P_i) concentration causes the change of defect behavior from V-type to I-type in HP-Cz-Si.

Furthermore, in the 10²⁰ P cm⁻³ order, we have identified even more peculiar defect behavior in the report (I).²⁷ It was experimentally found that, at 1.3 × 10²⁰ P cm⁻³, small dislocation loops were observed in the bottom, while interstitial-type stacking faults (SFs) with P segregation were observed in the middle of crystal. The growing and tangling dislocations and P segregation were also observed in the crystal portion for a longer thermal history around 600 °C. The P segregation suggests the existence of supersaturated interstitial P_i atoms. However, no quantitative explanation of P_i concentration during HP-Cz-Si crystal growth has been given. Furthermore, various points still need to be clarified, such as the formation of SFs, the expansion of dislocations, and the actual state of defects in low-temperature regions during crystal growth of HP-Cz-Si.

In the present paper, the thermal equilibrium concentrations of substitutional P (P_s), interstitial P (P_i), and (P_s)_nV (n = 1−4) clusters in Si are obtained on the basis of DFT calculations. We propose an appropriate model of intrinsic point defect behavior in growing HP-Cz-Si. Furthermore, we perform DFT calculations for P_i segregation on the SF and for the photoelectron spectra of P 1s measured by hard x-ray photoelectron spectroscopy (HAXPES) to explain the other experimental results in the report (I).²⁷ 

DFT calculations were carried out within the generalized gradient approximation (GGA)²⁸ for electron exchange and correlation using the CAmbridge Serial Total Energy Package (CASTEP) code.²⁹ Three-dimensional periodic boundary conditions were set with cubic supercells of 512 Si atoms to calculate the total energy of Si crystals containing P_s, P_i, and (P_s)_nV clusters. The cut-off energy of the plane waves was 340 eV. We carried out k-point sampling at the Γ point. Note that if the model contains one P atom, its concentration [P] = 1 × 10²⁰ P cm⁻³ is close to the actual concentration of the heavily P doped Si crystal. The formation energy of these P defects was obtained after geometry optimization.

Here, $E tot[ S i 511 − n ( P s ) n V]$ corresponds to the total energy of the cell including one $( P s ) nV$ cluster.

Due to the high calculation cost of the linear response method,³⁰ formation (vibration) entropies of P_s atom [S_f (P_s)] and P_i atom [S_f (P_i)] were calculated using a cubic supercell of 64 Si atoms. The change of formation entropy through (P_s)_nV cluster formation was not taken into consideration due to its low calculation accuracy of relatively large (P_s)_nV cluster sizes for a 64-Si atom model.

The thermal equilibrium concentrations of the P_s, P_i, and $( P s ) nV$ clusters were obtained on the basis of the calculated E_f and S_f. The detail procedure will be described in Sec. III A.

An interstitial-type Frank-loop stacking fault (SF) was modeled as shown in Fig. 3 and compared with the perfect Si model. The SF model and perfect model consist of 32 and 30 (111) layers, respectively, including 16 atoms in each layer. To obtain the energy reduction of P_i at the SF with reference to the Si bulk, formation energy [E_f (P_i)] of the P_i atom in Eq. (1a) was calculated by moving the position of the P_i atom from the bulk to the SF.

By adding the other P_i atoms to the P_i atom on the SF, the most stable (P_i)₂–(P_i)₇ clusters on the SF were obtained among the cluster structures considered. The energy reduction in P_i cluster growth on the SF was then calculated.

Here, C_Si = 5 × 10²² cm⁻³ is the site number of Si [= site number of P_s in Eq. (3a), site number of P_i in Eq. (3b), and site number of V in Eq. (3c)], k is the Boltzmann constant, and T is the temperature. The first row of Eq. (3b) indicates the Cp_i of P_i atoms up to the 5th nearest substituted P_s atom, and the second row of Eq. (3b) indicates the Cp_i of P_i atoms beyond the 5th nearest substituted P_s atom. Here, we assumed that E_f of the P_i farther than the 5th nearest is not affected by the P_s atom. ₄C_n in Eq. (3c) is the degeneracy number of n P_s atoms at the nearest V.

Here, we assumed that the concentration of I maintained the thermal equilibrium concentration as the supersaturated I atoms were absorbed by the SFs and/or dislocation loops as will be mentioned later in this section.

Here, we also assumed that the concentration of I maintained the thermal equilibrium concentration.

Figure 5 shows the thermal equilibrium concentrations, Cp_s of P_s [Eq. (4a)], Cp_i of P_i [Eq. (6)], and $C ( P s ) n V of ( P s ) nV$ [Eqs. (7a)–(7d)] as a function of temperature. For Cp_i, the values of [P_s] at 1 × 10¹⁹, 5 × 10¹⁹, 1 × 10²⁰, and 2 × 10²⁰ P_s cm⁻³ are shown. Here, we assumed that the melting temperature T of HP-Cz-Si is 1412 °C. By using the thermal equilibrium concentration at the melting temperature, we obtained the incorporated P defect concentrations, [P_s] of P_s, [P_i] of P_i, and $[ ( P s ) n V]$ of $( P s ) nV$ as functions of total P concentrations [P]^tot as shown in Fig. 6(a). Here, we calculated [P_i] by Eq. (4c), $[ ( P s ) n V]$ by Eq. (4e), and [P]^tot by Eq. (5) at the given [P_s]. We clarified that the main P defects incorporated at the solid/liquid interface are P_s and P_i in the [P]^tot range of 1 × 10¹⁹–2 × 10²⁰ P cm⁻³. Figure 6(b) shows the P defect concentrations, [P_s] of P_s, [P_i] of P_i, and $[ ( P s ) n V]$ of $( P s ) nV$ as functions of total P concentrations [P]^tot at T = 600 °C. Figure 6(c) shows P_s, P_i, and (P_s)₄V concentrations at [P]^tot = 1 × 10²⁰ P cm⁻³ as a function of temperature. The obtained expressions, Cp_s [Eq. (4a)], Cp_i [Eq. (6)], and $C ( P s ) n V$ [Eqs. (7a)–(7d)], will be very impactful to expand the application of numerical simulation¹⁸ to HP-Cz-Si crystal growth. Here, we use these data in the following discussion on the mechanism of defect behaviors in HP-Cz-Si crystal growth.

Table II summarizes the incorporated P_s and P_i concentration at the melting temperature with their supersaturated temperatures. The data in Table II can be used quantitatively to explain our experimental results in the report (I)²⁷ of the peculiar defect behavior around 10²⁰ P cm⁻³ mentioned in Sec. I.

At the melting temperature, P_i around 10¹⁷ cm⁻³ is incorporated at the solid/liquid interface by heavily P doping around 10²⁰ P cm⁻³. V becomes supersaturated at a concentration of about 10¹⁴–10¹⁵ V cm⁻³ after the pair recombination with I.^4,35 P_i atoms become supersaturated at a concentration of about 10¹⁷ P_i cm⁻³ around void formation temperatures (around 1100 °C).³⁶ Since the calculated diffusion barrier of P_i atom is very small (0.14 eV), the reaction P_i + V → P_s proceeds, and the observable voids no longer form during crystal growth. Note that P_i still remains at [P_i] ∼ 10¹⁷ P_i cm⁻³ after the reaction P_i + V → P_s.

During cooling of the crystal down to about 600 °C, the supersaturated P_i kicks out a lattice Si atom (P_i → P_s + I). The [P_i] is decreasing as can be seen from Fig. 6(c). The I becomes supersaturated and forms I-type dislocation loops and SFs.²⁷ The concentration of I maintained the thermal equilibrium concentration as the supersaturated I atoms were absorbed by the SFs and/or dislocation loops.

At temperatures of 600 °C and lower, P_s becomes supersaturated. As can be seen from Figs. 6(b) and 6(c), the (P_s)₄V concentration balanced to that of P_s is about 10¹⁹ P₄V cm⁻³ when [P]^tot = 10²⁰ Pcm⁻³. That is, a (P_s)₄V cluster forms from the reaction 4P_s → (P_s)₄V + I at 600 °C and lower. We will further discuss of the formation of (P_s)₄V in Sec. III C. The generated I is absorbed by the I-type dislocation loops, causing defect growth and tangle. In addition, the P_i formed by P_s + I → P_i segregates on SFs.²⁷ The clustering of P_i atoms on SF will be discussed in Sec. III B. The formation of (P_s)₄V drastically proceeds during long-time wafer annealing at 600 °C and lower.²⁷ 

Finally, we briefly discuss the case when [P]^tot is less than 5 × 10¹⁹ P cm⁻³. As shown in Table II, P_i becomes supersaturated when below the void formation temperatures (around 1100 °C). Therefore, supersaturated V forms voids and later the reaction P_i → P_s + I occurs during crystal growth. This is probably the reason that voids were experimentally observed at [P]^tot less than 5 × 10¹⁹ P cm⁻³.⁴ 

Figure 7 shows the most stable configuration of the P_i atom at the SF and the formation energy of the P_i atom with reference that on the SF. We found that the P_i atom becomes about 0.3 eV more stable when the P_i atom is trapped on the SF. The energy is not greatly reduced but it will be a sufficient driving force for trapping as the supersaturation of P_i increases as the temperature decreases during crystal growth.

Table III summarizes the calculated binding energies of the P 1s electron (E_b) for P_s, P_i, and (P_s)_nV clusters and energy shift ΔE_b from the value of P_s. The charge state determined by the energy band calculations is also shown. By comparing the calculated results with the experimental HAXPES results [P1, P2 (inactive) and P2 (active) peaks] in the report (I),²⁷ we concluded that the origin of the P1 peak should be P_s. The origin of the P2 peak (inactive), which becomes noticeable after a longer thermal history at 600 °C and lower and/or long-time wafer annealing at 600 °C and lower, should be (P_s)₄V. (P_s)₂V is not the origin of the P2 peak (inactive) as its concentration is very low as shown in Fig. 6(b).

The P2 peak (active) was also observed in the as-grown crystal.²⁷  Figure 10 shows two types of P₃ clusters, (P_s + P_i) + P_s(a) and (P_s + P_i) + P_s(b), which are responsible for the P2 peak (active). These two clusters are stable as the energy of (P_s + P_i) + P_s(a) is reduced by −0.94 eV and that of (P_s + P_i) + P_s(b) is reduced by −1.10 eV from the isolated one P_s and two P_i. Furthermore, they have a +1 charge and ΔE_b close to that of the experimental results. That is, the two clusters are possible origins of P2 (active) formed during crystal growth.

HP-Cz-Si crystals with a resistivity down to 0.6 mΩ cm ([P] = 1.3 × 10²⁰ P cm⁻³) are currently being manufactured for application for low-voltage power MOSFETs. Recent studies including our own report (I)²⁷ have revealed that HP-Cz-Si exhibits peculiar defect behaviors such as the formation of SFs and dislocations. The purpose of this paper is to clarify the causes of the defect behavior in HP-Cz-Si through theoretical calculations.

The thermal equilibrium concentrations of P_s, P_i, and (P_s)_nV (n = 1−4) clusters were determined by DFT calculations. Furthermore, equilibrium concentrations of P_i and (P_s)_nV balanced to the given P_s concentration were obtained as functions of the total P concentration and the temperature. On the basis of the calculated results those can quantitatively explain our experimental results in the report (I),²⁷ we proposed the following defect model to represent HP-Cz-Si crystal growth. At the melting temperature, P_i around 10¹⁷ cm⁻³ is incorporated at the solid/liquid interface by heavily P doping around 10²⁰ P cm⁻³. From 1100 to 600 °C, supersaturated P_i atom interacts with the Si atom to become P_s with the emission of I. The emitted I_s agglomerate and form SFs and dislocations. At temperatures of 600 °C and lower, supersaturated P_s becomes (P_s)₄V with the emission of I. The SFs and dislocations absorb I_s and become more complex defects. The supersaturated P_i also segregates on the defects. The formation of (P_s)₄V further proceeds during long-time wafer annealing at 600 °C and lower.

The other DFT calculations explained the P_i segregation on the SF. The calculated energy is reduced by about 0.8–2.0 eV for each P_i associated with the growth of a P_i cluster up to (P_i)₇. That is, if one P_i is trapped on the SF, the growth of the P_i cluster will proceed drastically.

Finally, the binding energies of the P 1s electron for P_s, P_i, and (P_s)_nV clusters were calculated and compared to the experimental HAXPES results [P1, P2 (inactive) and P2 (active) peaks] in the report (I).²⁷ We concluded that the origin of the P1 peak should be P_s, and the origin of the P2 peak (inactive), which becomes noticeable after longer thermal history at 600 °C and lower and/or long-time wafer annealing at 600 °C and lower, should be (P_s)₄V. Lastly, we proposed two structures of the P₃ cluster responsible for the P2 peak (active) observed in the as-grown crystal.

This work was partially supported by JST, CREST, Japan (Grant No. JPMJCR21C2).

The authors have no conflicts to disclose.

Koji Sueoka: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Supervision (lead); Validation (lead); Writing – original draft (lead); Writing – review & editing (lead). Yasuhito Narushima: Investigation (supporting); Methodology (supporting); Validation (supporting). Kazuhisa Torigoe: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal). Naoya Nonaka: Investigation (supporting); Methodology (supporting); Validation (supporting). Koutaro Koga: Investigation (supporting); Methodology (supporting); Validation (supporting). Toshiaki Ono: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal). Hiroshi Horie: Formal analysis (equal); Investigation (supporting); Methodology (supporting); Validation (supporting). Masataka Hourai: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.